2_975_jhade.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 4, No. 4, 2011, 330-339

ISSN 1307-5543 – www.ejpam.com

Coincidence & Fixed Points Of Nonexpansive Type Multi-Valued

& Single Valued Maps

Pankaj Kumar Jhade1,∗, A. S. Saluja2, Renu Kushwah3

1 Department of Mathematics, NRI Institute of Information Science & Technology, Bhopal, India-

462021
2 Department of Mathematics, JH Government College, Betul, India-460001
3 Department of Mathematics, NRI Institute of Information Science & Technology, Bhopal, India-

462021

Abstract. Fixed point theory of nonexpansive and nonexpansive type single and multivalued mappings

provides techniques for solving a variety of applied problems in mathematical sciences and engineer-

ing. In this paper we consider the existence of coincidences and fixed points of nonexpansive type

conditions satisfied by multivalued and single valued maps and prove some fixed point theorems for

nonexpansive type single and multivalued mappings.

2000 Mathematics Subject Classifications: 47H10, 54H25

Key Words and Phrases: Coincidence and fixed points; nonexpansive mappings; compatible map-

pings; T-orbitally complete; (T,f)-orbitally complete.

1. Introduction & Preliminaries

Throughout this paper let (X , d) be a metric space and H denotes the Housdorff (resp.

generalized Housdorff) metric on CB (X ) (resp. C L (X )) induced by the metric d , where

CB (X ) (resp. C L (X )) is the collection of all nonempty closed and bounded (resp. closed),

subsets of X . For these definitions one may refer [1, 3, 6, 7]. For y ∈ X and A⊂ X , d
�

y,A
�

will denote the ordinary distance between y and A.

A map T : X → X is said to be nonexpansive if d
�

T x , T y
�

≤ d
�

x , y
�

for all x , y ∈ X . Ćirić

[4] investigated a class of self maps T of X which satisfy the following nonexpansive type

condition:

d(T x , T y) ≤ a max{d(x , y), d(x , T x), d(y, T y),
d(x , T y) + d(y, T x)

2
}

+ b max{d(x , T x), d(y, T y)}+ c[d(x , T y) + d(y, T x)]

(1)

∗Corresponding author.

Email addresses: pmathsjhade�gmail.
om (P. Jhade), dssaluja�rediffmail.
om (A. Saluja)

http://www.ejpam.com 330 c© 2011 EJPAM All rights reserved.



P. Jhade, A. Saluja, R. Kushwah / Eur. J. Pure Appl. Math, 4 (2011), 330-339 331

For all x , y ∈ X , where a, b, c ≥ 0 such that a+ b+ 2c = 1.

M. Chandra et al [2] consider the following generalization of (1), let T, f : X → X satisfy-

ing:

d(T x , T y)≤ a(x , y)d( f x , f y) + b(x , y)max{d( f x , T x), d( f y, T y)}

+ c(x , y)[d( f x , T y) + d( f y, T x)]
(2)

where, a(x , y)≥ 0, β = infx ,y∈X b(x , y)> 0, γ= infx ,y∈X c(x , y) > 0 and

supx ,y∈X [a(x , y) + b(x , y) + 2c(x , y)] = 1 and prove some fixed point theorems for single

valued and multi valued maps. They also prove that (1) contained in (2).

In this paper we use the following nonexpansive type condition: Let T, f : X → X be two

self mappings satisfying the condition,

d(T x , T y)≤ a(x , y)d( f x , f y) + b(x , y)max{d( f x , T x), d( f y, T y)}

+ c(x , y)max{d( f x , f y), d( f x , T x), d( f y, T y)}

+ e(x , y)max{d( f x , f y), d( f x , T x), d( f y, T y), d( f x , T y)}

(3)

Where a(x , y), b(x , y), c(x , y), e(x , y) ≥ 0 and β = infx ,y∈X e(x , y)> 0,

γ= infx ,y∈X (1+b(x , y)+e(x , y))> 0 with supx ,y∈X (a(x , y)+b(x , y)+c(x , y)+2e(x , y)) = 1.

Definition 1 ([5]). Let f and g be two self maps of a metric space X . Then f and g are

said to be compatible if limn→∞ d( f g xn, g f xn) = 0, whenever {xn} is a sequence such that

limn→∞ f xn = limn→∞ g xn = t ∈ X .

2. Main Results

Theorem 1. Let (X , d) be a metric space, T, f are self maps of X satisfying (3) with T (X )⊆ f (X )

and either (a) X is complete and f is surjective; or (b) X is complete, f is continuous and T, f

are compatible; or (c) f (X ) is complete ; or (d) T (X ) is complete. Then f and T have a coinci-

dence point in X . Further, the coincidence value is unique, i.e. f p = f q whenever f p = T p and

f q = Tq (p,q ∈ X ).

Proof. Let x0 ∈ X . Since T (X ) ⊆ f (X ), choose x1 so that y1 = f x1 = T x0. In general,

choose xn+1 such that yn+1 = f xn+1 = T xn.

From (3), we have

d(T xn, T xn+1)≤ ad( f xn, f xn+1) + b max{d( f xn, T xn), d( f xn+1, T xn+1)}

+ c max{d( f xn, f xn+1), d( f xn, T xn), d( f xn+1, T xn+1)}

+ e max{d( f xn, f xn+1), d( f xn, T xn), d( f xn+1, T xn+1), d( f xn, T xn+1)}

≤ ad( f xn, T xn)+ b max{d( f xn, T xn), d( f xn+1, T xn+1)}

+ c max{d( f xn, T xn), d( f xn, T xn), d( f xn+1, T xn+1)}

+ e max{d( f xn, T xn), d( f xn, T xn), d( f xn+1, T xn+1), d( f xn, T xn) + d( f xn+1, T xn+1)}



P. Jhade, A. Saluja, R. Kushwah / Eur. J. Pure Appl. Math, 4 (2011), 330-339 332

where a, b, c and e are evaluated at (xn, xn+1). Suppose that for some n,

d( f xn+1, T xn+1)> d( f xn, T xn). Then substituting in the above inequality we have

d( f xn+1, T xn+1)< (a+ b+ c + 2e)d( f xn+1, T xn+1)

a contradiction. Therefore, for all n we have

d( f xn+1, T xn+1)≤ d( f xn, T xn) (4)

Again d(yn−1, T xn) = d(T xn−2, T xn)

Using (3), (4) and triangle inequality we have

d(yn−1, T xn)≤ ad( f xn−2, f xn) + b max{d( f xn−2, T xn−2), d( f xn, T xn)}

+ c max{d( f xn−2, f xn), d( f xn−2, T xn−2), d( f xn, T xn)}

+ e max{d( f xn−2, f xn), d( f xn−2, T xn−2), d( f xn, T xn), d( f xn−2, T xn)}

≤ 2ad( f xn−2, T xn−2) + bd( f xn−2, T xn−2) + 2cd( f xn−2, T xn−2)

+ e max{2d( f xn−2, T xn−2), d( f xn−2, T xn−2) + d( f xn−1, T xn)}

≤ (2a+ b+ 2c + 3e)d( f xn−2, T xn−2)

implies that

d(yn−1, T xn)≤ (1− b− e)d( f xn−2, T xn−2) (5)

Using (3), (4) and (5) we obtain,

d(yn, T xn) = d(T xn−1, T xn)

≤ ad( f xn−1, f xn) + b max{d( f xn−1, T xn−1), d( f xn, T xn)}

+ c max{d( f xn−1, f xn), d( f xn−1, T xn−1), d( f xn, T xn)}

+ e max{d( f xn−1, f xn), d( f xn−1, T xn−1), d( f xn, T xn), d( f xn−1, T xn)}

≤ ad( f xn−2, T xn−2) + bd( f xn−2, T xn−2) + cd( f xn−2, T xn−2)

+ e(1− b− e)d( f xn−2, T xn−2)

≤ (1− e(1+ b+ e))d( f xn−2, T xn−2)

≤ (1− βγ)d( f xn−2, T xn−2)

≤ (1− βγ)
n/2d(y0, y1)

where β = infx ,y∈X e(x , y) > 0, γ = infx ,y∈X (1+ b(x , y) + e(x , y)) > 0 and {yn} is Cauchy,

hence converges to a point p in X .

Case (a):- Suppose that f is surjective. Then there exists a point z in X such that p = f z.

From (3), we have



P. Jhade, A. Saluja, R. Kushwah / Eur. J. Pure Appl. Math, 4 (2011), 330-339 333

d( f z, Tz) ≤ d( f z, yn+1) + d(yn+1, Tz)

≤ d( f z, yn+1) + ad( f xn, f z) + b max{d( f xn, T xn), d( f z, Tz)}

+ c max{( f xn, f z), d( f xn, T xn), d( f z, Tz)}

+ e max{d( f xn, f z), d( f xn, T xn), d( f z, Tz), d( f xn, Tz)}

≤ d( f z, f xn+1) + sup
x ,y∈X

(b+ c + e)max{max{d( f xn, T xn), d( f z, Tz)}

max{d( f xn, f z), d( f xn, T xn), d( f z, Tz)},

max{d( f xn, f z), d( f xn, T xn), d( f z, Tz), d( f xn, Tz)}}

+ sup
x ,y∈X

ad( f xn, f z)

Taking limit as n→∞, we get d( f z, Tz) ≤ supx ,y∈X (b+ c+ e)d( f z, Tz) implies that f z = Tz.

Case (b): Suppose f is continuous and f and T are compatible. Then since

limn→∞ yn = p, we have limn→∞ f yn = f p.

Now,

d( f p, T p) ≤ d( f p, f yn+1) + d( f yn+1, T p)

≤ d( f p, f yn+1) + d( f T xn, T f xn) + d(T f xn, T p)

Note that since limn→∞ f xn = limn→∞ T xn and f , T are compatible,

limn→∞ d
�

f T xn, T f xn

�

= 0.

From (3),we have

d(T f xn, T p) ≤ ad( f f xn, f p) + b max{d( f f xn, T f xn), d( f p, T p)}

+ c max{d( f f xn, f p), d( f f xn, T f xn), d( f p, T p)}

+ e max{d( f f xn, f p), d( f f xn, T f xn), d( f p, T p), d( f f xn, T p)}

≤ sup
x ,y∈X

a(x , y)d( f f xn, f p) + sup
x ,y∈X

(b(x , y) + c(x , y) + e(x , y))

max{max{d( f f xn, T f xn), d( f p, T p)},

max{d( f f xn, f p), d( f f xn, T f xn), d( f p, T p)},

max{d( f f xn, f p), d( f f xn, T f xn), d( f p, T p), d( f f xn, T p)}}

Note that d( f f xn, T f xn) ≤ d( f f xn, f T xn) + d( f T xn, T f xn). Using the continuity of f and

compatibility of f and T , it follows that lim d( f f xn, T f xn) = 0. Since lim f f xn = f p, it

follows that lim T f xn = f p.

Substituting into the above inequality and taking limit as n→∞, we get

d( f p, T p) ≤ supx ,y∈X (b(x , y)+ c(x , y) + e(x , y))d( f p, T p) implies that f p = T p.

Case (c): In this case p ∈ f (X ). Let z ∈ f −1p. Then p = f z and the proof is complete by

case (a).



P. Jhade, A. Saluja, R. Kushwah / Eur. J. Pure Appl. Math, 4 (2011), 330-339 334

Case (d): In this case p ∈ T (X )⊆ f (X ) and the proof is complete by case (c).

Uniqueness: Let q be another coincidence point of f and T , then from (3) with a, b, c

and d evaluated at (p,q),

d(T p, Tq) ≤ ad( f p, f q) + b max{d( f p, T p), d( f q, Tq)}

+ c max{d( f p, f q), d( f p, T p), d( f q, Tq)}

+ e max{d( f p, f q), d( f p, T p), d( f q, Tq), d( f p, Tq)}

≤ (a+ c + e)d(T p, Tq)

This implies that T p = Tq and hence f p = f q.

Corollary 1. Let (X , d) be a complete metric space and T a self mapping of X satisfying (3) with

f = I , the identity map on X and supx ,y∈X (a(x , y) + 2b(x , y) + c(x , y) + e(x , y)) = 1 . Then

T has a unique fixed point and at this fixed point T is continuous.

Proof. The existence and uniqueness of the fixed point comes from Theorem 1 by setting

f = I . To prove continuity , let {yn} ⊂ X with lim yn = p, p the unique fixed point of T .

Using (3), we have

d(T yn, T p) ≤ ad(yn, p) + b max{d(yn, T yn), d(p, T p)}

+ c max{d(yn, p), d(yn, T yn), d(p, T p)}

+ e max{d(yn, p), d(yn, T yn), d(p, T p), d(yn, T p)}

≤ ad(yn, p) + b max{d(yn, T yn), d(p, p}

+ c max{d(yn, p), d(yn, T yn), d(p, p)}

+ e max{d(yn, p), d(yn, T yn), d(p, p), d(yn, p)}

≤ ad(yn, p) + b[d(yn, p) + d(p, T yn)]

+ c max{d(yn, p), d(yn, p) + d(p, T yn)}

+ e max{d(yn, p), d(yn, p) + d(p, T yn)}

Hence

d(T yn, p) ≤ sup
x ,y∈X

(a+ b)d(yn, p) + sup
x ,y∈X

(b+ c + e)

max{d(p, T yn),max{d(yn, p), d(yn, p) + d(p, T yn)}}

≤ sup
x ,y∈X

(
b+ c + e

1− a− b
)d(yn, p)

Taking limit as n→∞ we get lim T yn = p = T p.

Next we establish some results when T is a multi-valued map from a metric space X to the

collection of nonempty subset of X , and f is a self map of X . Let C(X ) denote the collection

of all nonempty compact subset of X .



P. Jhade, A. Saluja, R. Kushwah / Eur. J. Pure Appl. Math, 4 (2011), 330-339 335

Definition 2 ([3]). An orbit of the multi-valued map T at a point x0 in X is a sequence

{xn : xn ∈ T xn−1}. A space X is T−orbitally complete if every Cauchy sequence of the form

{xni
: xni
∈ T xni−1} converges in X .

Definition 3 ([7]). If for a point x0 in X , there exists a sequence {xn} ⊂ X such that

f xn+1 ∈ T xn, n = 0,1,2, · · · , then Of (x0) = { f xn : n = 1,2, · · · } is an orbit of (T, f ) at x0. A

space X is called (T, f )−orbitally complete if every Cauchy sequence of the form

{ f xni
: f xni

∈ T xni−1} converges in X .

Theorem 2. Let X be a metric space, T a multi-valued map from X to C(X ). Let f be a self map

of X such that T (X )⊆ f (X ) and either of the following conditions is satisfied:

1. X is (T, f )−orbitally complete and f is surjective;

2. f (X ) is (T, f )−orbitally complete;

3. T (X ) is (T, f )−orbitally complete.

Suppose that T and f satisfy the condition:

H(T x , T y) ≤ a(x , y)d( f x , f y) + b(x , y)max{d( f x , T x), d( f y, T y)}

+ c(x , y)max{d( f x , f y), d( f x , T x), d( f y, T y)}

+ e(x , y)max{d( f x , f y), d( f x , T x), d( f y, T y), d( f x , T y)}

(6)

Where a, b, c and e are nonnegative functions from X×X → [0,1) such that infx ,y∈X e(x , y) > 0,

infx ,y∈X (1+ b(x , y)+ e(x , y))> 0 and supx ,y∈X (a+ b+ c+2e)(x , y) = 1. Then f and T have

a coincidence, i.e. there exists a point z in X such that f z ∈ Tz.

Proof. Choose x0 ∈ X . We construct sequences {xn} and {yn} as follows: Since

T (X ) ⊆ f (X ), we can choose y1 = f x1 ∈ T x0. If T x0 = T x1, Choose y2 = f x2 ∈ T x1 such

that y1 = y2. If T x0 6= T x1, choose y2 = f x2 ∈ T x1 such that d(y1, y2)≤ H(T x0, T x1). Such

a choice is possible since T x is compact for each x in X . In general, choose

yn+2 = f xn+2 ∈ T xn+1 such that yn+1 = yn+2 if T xn = T xn+1 and d(yn+1, yn+2)≤ H(T xn, T xn+1)

otherwise. From (6) with a, b, c and e evaluated at (xn, xn+1),

d(yn+1, yn+2)≤ H(T xn, T xn+1)

≤ ad( f xn, f xn+1) + b max{d( f xn, T xn), d( f xn+1, T xn+1)}

+ c max{d( f xn, f xn+1), d( f xn, T xn), d( f xn+1, T xn+1)}

+ e max{d( f xn, f xn+1), d( f xn, T xn), d( f xn+1, T xn+1), d( f xn, T xn+1)}

≤ ad(yn, yn+1)+ b max{d(yn, yn+1), d(yn+1, yn+2)}

+ c max{d(yn, yn+1), d(yn, yn+1), d(yn+1, yn+2)}

+ e max{d(yn, yn+1), d(yn, yn+1), d(yn+1, yn+2), d(yn, yn+2)}

If for some n, d(yn+1, yn+2)> d(yn, yn+1), the above inequality gives

d(yn+1, yn+2)< (a+ b+ c + 2e)d(yn+1, yn+2)



P. Jhade, A. Saluja, R. Kushwah / Eur. J. Pure Appl. Math, 4 (2011), 330-339 336

a contradiction. Therefore, for each n, we have

d(yn+1, yn+2)≤ d(yn, yn+1) (7)

Again from (6) with a, b, c and e evaluated at (xn−2, xn)

d(yn−1, T xn)≤ H(T xn−2, T xn)

≤ ad( f xn−2, f xn) + b max{d( f xn−2, T xn−2), d( f xn, T xn)}

+ c max{d( f xn−2, f xn), d( f xn−2, T xn−2), d( f xn, T xn)}

+ e max{d( f xn−2, f xn), d( f xn−2, T xn−2), d( f xn, T xn), d( f xn−2, T xn)}

Using (7) and triangle inequality we get

d(yn−1, T xn)≤ 2ad(yn−2, yn−1) + bd(yn−2, yn−1) + 2cd(yn−2, yn−1)

+ e max{2d(yn−2, yn−1), d(yn−2, yn−1) + d( f xn−1, T xn−1) + d( f xn, T xn)}

+ (2a+ b+ 2c)d(yn−2, yn−1) + e max{2d(yn−2, yn−1), 3d(yn−2, yn−1)}

+ (2a+ b+ 2c+ 2e)d(yn−2, yn−1)

Implies that

d(yn−1, T xn)≤ (1− b− e)d(yn−1, yn−2) (8)

Again from (6), (7) and using (8)

d(yn, yn+1) = d( f xn, f xn+1)

≤ H(T xn−1, T xn)

≤ ad( f xn−1, f xn) + b max{d( f xn−1, T xn−1), d( f xn, T xn)}

+ c max{d( f xn−1, f xn), d( f xn−1, T xn−1), d( f xn, T xn)}

+ e max{d( f xn−1, f xn), d( f xn−1, T xn−1), d( f xn, T xn), d( f xn−1, T xn)}

≤ ad( f xn−2, T xn−2) + bd( f xn−2, T xn−2) + cd( f xn−2, T xn−2)

+ e(1− b− e)d( f xn−2, T xn−2)

≤ [1− e(1+ b+ e)]d( f xn−2, T xn−2)

≤ (1− βγ)d(yn−2, yn−1)

≤ (1− βγ)
n/2d(y0, y1)

and
�

yn

	

is Cauchy, hence convergent to a point p in X in Cases (i)-(iii).

If f is surjective, there exists a point z such that p = f z. This is obviously true in cases (ii)



P. Jhade, A. Saluja, R. Kushwah / Eur. J. Pure Appl. Math, 4 (2011), 330-339 337

and (iii) as well,

d( f z, Tz) ≤ d( f z, f xn+1) + d( f xn+1, Tz)

≤ d( f z, f xn+1) +H(T xn, Tz)

≤ d( f z, f xn+1) + ad( f xn, f z)

+ b max{d( f xn, T xn), d( f z, Tz)}

+ c max{d( f xn, f z, d( f xn, T xn), d( f z, Tz)}

+ e max{d( f xn, f z), d( f xn, T xn), d( f z, Tz), d( f xn, Tz)}

≤ d( f z, f xn+1) + sup
x ,y∈X

ad( f xn, f z)

+ sup
x ,y∈X

(b+ c + e)max{max{d( f xn, T xn), d( f z, Tz)},

max{d( f xn, f z), d( f xn, T xn), d( f z, Tz)},

max{d( f xn, f z), d( f xn, T xn), d( f z, Tz), d( f xn, Tz)}}

Taking limit as n→∞

d( f z, Tz) ≤ d( f z, f z) + sup
x ,y∈X

ad( f z, f z)

+ sup
x ,y∈X

(b+ c + e)max{max{d( f z, Tz), d( f z, Tz)}

,max{d( f z, f z), d( f z, Tz), d( f z, Tz)},max{d( f z, f z), d( f z, Tz), d( f z, Tz), d( f z, Tz)}}

Implies that d( f z, Tz) ≤ supx ,y∈X (b+ c + e)d( f z, Tz)and hence f z = Tz.

Corollary 2. Let X be a metric space, T a multi-valued map from X to C(X ). If X is T -orbitally

complete and for each x , y ∈ X

H(T x , T y) ≤ a(x , y)d(x , y) + b(x , y)max{d(x , T x), d(y, T y)}

+ c(x , y)max{d(x , y), d(x , T x), d(y, T y)}

+ e(x , y)max{d(x , y), d(x , T x), d(y, T y), d(x , T y)}

(9)

Where a, b, c, e : X × X → [0,1) satisfying a(x , y)≥ 0, infx ,y∈X e(x , y)> 0

infx ,y∈X (1+ b(x , y) + e(x , y))> 0 and sup
x ,y∈X

(a(x , y) + b(x , y) + c(x , y) + 2e(x , y)) = 1

Then T has a fixed point in X.

Theorem 3. Let X , T and f satisfy the hypotheses of Theorem 2 with C(X ) replaced by C L(X )

and a, b, c, e satisfy δ = supx ,y∈X (a(x , y) + b(x , y) + c(x , y) + 2e(x , y) < 1. Then T and f

have a coincidence.

Proof. Let x0 ∈ X , and construct sequences {xn} and {yn} as follows: since T (X ) ⊆ f (X ),

choose y1 = f x1 ∈ T x0. If T x0 = T x1 choose y2 = f x2 ∈ T x1 such that y1 = y2. If



P. Jhade, A. Saluja, R. Kushwah / Eur. J. Pure Appl. Math, 4 (2011), 330-339 338

T x0 6= T x1, choose y2 = f x2 ∈ T x1 such that d(y1, y2) ≤ λH(T x0, T x1), where λ > 1 and

λδ < 1. In general, choose yn+2 ∈ T xn+1 such that d(yn+1, yn+2) ≤ λH(T xn, T xn+1). From

(6), we have

d(yn+1, yn+2) = d( f xn+1, f xn+2)

≤ H(T xn, T xn+1)

≤ λa(x , y)d( f xn, f xn+1) +λb(x , y)max{d( f xn, T xn), d( f xn+1, T xn+1)}

+λc(x , y)max{d( f xn, f xn+1), d( f xn, T xn), d( f xn+1, T xn+1)}

+λe(x , y)max{d( f xn, f xn+1), d( f xn, T xn), d( f xn+1, T xn+1), d( f xn, T xn+1)}

≤ λad(yn, yn+1) +λb max{d(yn, yn+1), d(yn+1, yn+2)}

+λc max{d(yn, yn+1), d(yn, yn+1), d(yn+1, yn+2)}

+λe max{d(yn, yn+1), d(yn, yn+1), d(yn+1, yn+2), d(yn, yn+2)}

If there exists an n such that d(yn+1, yn+2)> d(yn, yn+1), we have

d(yn+1, yn+2)< λ(a+ b+ c + 2e)d(yn+1, yn+2)

a contradiction. Therefore for all n we get d
�

yn+1, yn+2

�

≤ d
�

yn, yn+1

�

and we obtain

d(yn+1, yn+2)≤ λad(yn, yn+1)+λbd(yn, yn+1) +λcd(yn, yn+1) + 2λed(yn, yn+1)

≤ λ(a+ b+ c + 2e)d(yn, yn+1)

≤ kd(yn, yn+1)≤ knd(y0, y1) where k = sup
x ,y∈X

λ(a+ b+ c + 2e)

Therefore {yn} is Cauchy, hence convergent to some point p in X . Since f is surjective, there

exists a z such that f z = p.

Now

d( f z, Tz) ≤ d( f z, f xn+1) + d( f xn+1, Tz)

≤ d( f z, f xn+1) +H(T xn, Tz)

≤ d( f z, f xn+1) + ad( f xn, f z)

+ b max{d( f xn, T xn), d( f z, Tz)}

+ c max{d( f xn, f z), d( f xn, T xn), d( f z, Tz)}

+ e max{d( f xn, f z), d( f xn, T xn), d( f z, Tz), d( f xn, Tz)}

≤ d( f z, f xn+1) + sup
x ,y∈X

ad( f xn, f z)

+ sup
x ,y∈X

(b+ c + e)max{max{d( f xn, T xn), d( f z, Tz)},

max{d( f xn, f z), d( f xn, T xn), d( f z, Tz)},

max{d( f xn, f z), d( f xn, T xn), d( f z, Tz), d( f xn, Tz)}}

Taking limit as n→∞ we get d
�

f z, Tz
�

≤ supx ,y∈X (b+ c + e) d
�

f z, Tz
�

implies that

f z ∈ Tz.



REFERENCES 339

References

[1] R. Chandel and A. Ganguly, Bull. Calcutta Math. Soc., 32:193-198,1990.

[2] M. Chandra, S. Mishra, S. Singh and B. Rhoades, Coincidence and fixed points

of nonexpansive type multi-valued and single-valued maps, Indian J. Pure Appl.

Math.,26(5):393-401,1995.

[3] Lj. Ćirić, Fixed points for generalized multi-valued contractions, Mat. Vesnik , 9:265-

272,1972.

[4] Lj. Ćirić, On some nonexpansive type mappings and fixed points, Indian J. Pure Appl.

Math., 24(3):145-149,1993.

[5] G. Jungck, Compatible mappings and fixed points, Int. J. Math. & Math. Sci.,9(4):771-

779,1986.

[6] H. Kaneko and S. Sessa, Fixed point theorem for compatible multi-valued and single-

valued mappings, Int. J. Math. & Math. Sci., 12(2): 257-262,1989.

[7] B. Rhoades, S. Singh and C. Kulshrestha, Coincidence theorems for some multi-valued

mappings, Int. J. Math. & Math. Sci.,7:429-434,1984.